
For any real numbers \(k\ge 0\)  and \(p\in [1,\infty ]\) , we set
\(\mathcal {L}^p := L^p(\Omega )\times L^p(\Gamma ),\quad \text{and}\quad \mathcal {H}^k := H^k(\Omega ) \times H^k(\Gamma ),\) 

and we identify \(\mathcal {L}^2\)  with \(\mathcal {H}^0\) . Note that \(\mathcal {H}^k\)  is a Hilbert space with respect to the inner product
\(\big ( (\phi ,\psi ) , (\zeta ,\xi ) \big )_{\mathcal {H}^k} := \big ( \phi  , \zeta  \big )_{H^k(\Omega )} + \big ( \psi  , \xi  \big )_{H^k(\Gamma )}\quad \text{for all $(\phi ,\psi ),(\zeta ,\xi )\in \mathcal {H}^k$,}\) 

and its induced norm \(\Vert  \cdot  \Vert _{\mathcal {H}^k}:=  \left( \cdot  , \cdot  \right)_{\mathcal {H}^k}^{1/2}\) .


For \(\beta >0\) , we introduce the subspace
\({\mathcal {D}_{\beta }}:= \left\lbrace  (\phi ,\psi ) \in \mathcal {H}^1 \;\big \vert \; \phi \vert _\Gamma = \beta \psi \;\; \text{a.e. on}\; \Gamma \right\rbrace  \subset \mathcal {H}^1,\) 

endowed with the inner product
\(  \left( \cdot  , \cdot  \right)_{\mathcal {D}_{\beta }}:=  \left( \cdot  , \cdot  \right)_{\mathcal {H}^1}\) 
and its induced norm.
The space \({\mathcal {D}_{\beta }}\)  is a Hilbert space. Moreover, we define the product
\( \big < (\phi ,\psi ) , (\zeta ,\xi ) \big >_{{\mathcal {D}_{\beta }}} :=  \left( \phi  , \zeta  \right)_{L^2(\Omega )} +  \left( \psi  , \xi  \right)_{L^2(\Gamma )}\) 

for all \((\phi ,\psi ), (\zeta ,\xi )\in \mathcal {L}^2\) . By means of the Riesz representation theorem, this product can be extended to a duality pairing on \({\mathcal {D}_{\beta }^{\prime }}\times {\mathcal {D}_{\beta }}\) , which will also be denoted as \( \langle \cdot  , \cdot  \rangle _{{\mathcal {D}_{\beta }}}\) .
In particular, the spaces \(\big ({\mathcal {D}_{\beta }},\mathcal {L}^2,{\mathcal {D}_{\beta }^{\prime }}\big )\)  form a Gelfand triplet, and the operator norm on \({\mathcal {D}_{\beta }^{\prime }}\)  is given by
\(\Vert  (\phi ,\psi ) \Vert _{{\mathcal {D}_{\beta }^{\prime }}} := \sup \Big \lbrace \, \big | \langle (\phi ,\psi ) , (\zeta ,\xi ) \rangle _{{\mathcal {D}_{\beta }}}\big | \;\Big \vert \; (\zeta ,\xi )\in {\mathcal {D}_{\beta }}\text{ with } \Vert  (\zeta ,\xi ) \Vert _{{\mathcal {D}_{\beta }}} = 1 \Big \rbrace ,\) 

for all \((\phi ,\psi ) \in {\mathcal {D}_{\beta }^{\prime }}\) .


We further recall the following bulk-surface Poincaré inequalities:


For any \(\alpha >0\) , there exists a constant \(C_P\)  depending only on \(\alpha \)  and \(\Omega \)  such that
\(\Vert  (\phi ,\psi ) \Vert _{\mathcal {L}^2}\le C_P \Vert  (\nabla \phi ,\nabla _\Gamma \psi ) \Vert _{\mathcal {L}^2}\) 
for all \((\phi ,\psi )\in {\mathcal {D}_{\alpha }}\)  with
\(\alpha \left| \Omega  \right|\langle \phi  \rangle _\Omega + \left| \Gamma  \right|\langle \psi  \rangle _\Gamma = 0\) .
This bulk-surface Poincaré inequality is a special case of the one established in [1]} (with the parameters therein being chosen as \(K=0\)  and \(\alpha ,\beta >0\) ).


There exists a constant \(C_P^{\prime }\)  depending only on \(\Omega \)  such that
\(\Vert  u \Vert _{L^2(\Omega )}\le C_P^{\prime } \left( \Vert  \nabla u \Vert _{L^2(\Omega )}+ \Vert  u \Vert _{L^2(\Gamma )} \right)\) 
for all \(u \in H^1(\Omega )\) .
This bulk-surface Poincaré inequality is established in [2]}.


